a.(1pt) Write down the equation for discriminant function for region 2 as specified by QDA, using values computed from the data. (Use standardised variables.) Also, explain what assumption is not satisfied for QDA to be correctly applied for this data.

Under the QDA rule, the following equation are for discriminant function

FOr the formula of y

According to the data output showed below, we can get the discriminant function for region 2 is

where in the following output we can get that

will be

$$

Under the QDA rule, the data should be a equal variance-covariance. In this question , It’s an unequal variance and covariance, so here we can conclude that the rule is not suitable in this case.

Moreover, the distribution of the observation in each class shld be a normal distribution with a class mean and class ocvariance. As for this question the 2 variables are not normal distribution instead of a multimodal distribtion. Thus, olive data cannot satisfied the assumption

# Call:
# qda(region ~ ., data = olive_train)
# 
# Prior probabilities of groups:
#         2         3 
# 0.3952096 0.6047904 
# 
# Group means:
#    arachidic   linoleic
# 2  0.7483237  1.0746306
# 3 -0.4893105 -0.6861894
#              arachidic     linoleic
# arachidic  0.178646453 -0.001300898
# linoleic  -0.001300898  0.169523093

b.(1pt) Make similar plots that shows the boundary between the two regions, that would be produced by .

(i) linear discriminant analysis,

# parsnip model object
# 
# Fit time:  21ms 
# Call:
# lda(region ~ ., data = data)
# 
# Prior probabilities of groups:
#         2         3 
# 0.3952096 0.6047904 
# 
# Group means:
#    arachidic   linoleic
# 2  0.7483237  1.0746306
# 3 -0.4893105 -0.6861894
# 
# Coefficients of linear discriminants:
#                  LD1
# arachidic -0.9182956
# linoleic  -2.0375057

This plot below have already showed the Boundary under the LDA prediction method:

In this question, we plot the boundary under training data for the reason that the training data have more observations compared to the test data, and the boundary under training data is also more precise than the boundary in test data.

(ii)* classification tree (using minsplit of 10) using the rpart engine.

Here we use ‘rpart’ engine to construct the tree, from the following table,we can conclude that there rare 167 roots, there are totally 2 classes are separated. In the class 2 , 66 roots are put in and selection criteria is linoleic>=0.5366322.In class 3, there are totally 101 roots are selected in and criteria is linoleic< 0.5366322.

# n= 167 
# 
# node), split, n, loss, yval, (yprob)
#       * denotes terminal node
# 
# 1) root 167 66 3 (0.3952096 0.6047904)  
#   2) linoleic>=0.5366322 66  0 2 (1.0000000 0.0000000) *
#   3) linoleic< 0.5366322 101  0 3 (0.0000000 1.0000000) *

In this graph below, we can tell there is a green line made under Classification Tree Method when we use the rpart engine and well separated 2 datasets.

c. (1pt) Compute the test balanced accuracy for the three models (i) LDA,(ii) QDA, (iii) classification tree. Are they equally as accurate?

Below are the degree of freedom of 3 methods

#    truth predict_lda predict_tree predict_qda
# 33     3           3            3           3
# 34     3           3            3           3
# 35     3           3            3           3
# 36     3           3            3           3
# 37     3           3            3           3
# 38     3           3            3           3
# 39     3           3            3           3
# 40     3           3            3           3
# 41     3           3            3           3
# 42     3           3            3           3
# 43     3           3            3           3
# 44     3           3            3           3
# 45     3           3            3           3
# 46     3           3            3           3
# 47     3           3            3           3
# 48     3           3            3           3
# 49     3           3            3           3
# 50     3           3            3           3
# 51     3           3            3           3
# 52     3           3            3           3
# 53     3           3            3           3
# 54     3           3            3           3
# 55     3           3            3           3
# 56     3           3            3           3
# 57     3           3            3           3
# 58     3           3            3           3
# 59     3           3            3           3
# 60     3           3            3           3
# 61     3           3            3           3
# 62     3           3            3           3
# 63     3           3            3           3
# 64     3           3            3           3
# 65     3           3            3           3
# 66     3           3            3           3
# 67     3           3            3           3
# 68     3           3            3           3
# 69     3           3            3           3
# 70     3           3            3           3
# 71     3           3            3           3
# 72     3           3            3           3
# 73     3           3            3           3
# 74     3           3            3           3
# 75     3           3            3           3
# 76     3           3            3           3
# 77     3           3            3           3
# 78     3           3            3           3
# 79     3           3            3           3
# 80     3           3            3           3
# 81     3           3            3           3
# 82     3           3            3           3

Here we can show that there are totally rows of 32:

# [1] 32

Then we calculate the balanced accuracy as follows:

# [1] 1
# [1] 1
# [1] 1

All of them are equally as accurate. The

this means all of these 3 models have performed well in seperating different classes, as the balanced accuracy for each model are equal to 1 (100%) .

d. (1pt) Re-fit the three models with the additional variable oleic (so you now have three predictors). Write a paragraph discussing how the models change, and why this might be. ?

Here, first of all we should take a look of the confusion matrix of all the 3 models.

Here we reform the LDA model with oleic as follows:

# Call:
# lda(region ~ arachidic + linoleic, data = olive_train1)
# 
# Prior probabilities of groups:
#         2         3 
# 0.3952096 0.6047904 
# 
# Group means:
#    arachidic   linoleic
# 2  0.7483237  1.0746306
# 3 -0.4893105 -0.6861894
# 
# Coefficients of linear discriminants:
#                  LD1
# arachidic -0.9182956
# linoleic  -2.0375057

We also reform the QDA model with variable oleic

# Call:
# qda(region ~ arachidic + linoleic, data = olive_train1)
# 
# Prior probabilities of groups:
#         2         3 
# 0.3952096 0.6047904 
# 
# Group means:
#    arachidic   linoleic
# 2  0.7483237  1.0746306
# 3 -0.4893105 -0.6861894

We reform tree model after we added the oleic

# n= 167 
# 
# node), split, n, loss, yval, (yprob)
#       * denotes terminal node
# 
# 1) root 167 66 3 (0.3952096 0.6047904)  
#   2) linoleic>=0.5366322 66  0 2 (1.0000000 0.0000000) *
#   3) linoleic< 0.5366322 101  0 3 (0.0000000 1.0000000) *

The matrix for LDA

#           Truth
# Prediction  2  3
#          2 32  0
#          3  0 50

Matrix for QDA

#           Truth
# Prediction  2  3
#          2 31  0
#          3  1 50

Here, in comparison with the 3 models, based on balance accuracy, we can get that after added the other variable ‘oleic’. The precision of LDA and tree model are same as without variable ‘oleic’. This means that after adding an extra variable, the LDA and tree model are still accurate to estimate this new model. However, the accuracy of QDA is changed from 1 to 0.984375. Although this accuracy is still high, however, in comparison with the previous one, adding an extra variable made the balance accuracy of QDA model decreased by 1-0.984375=0.015626

a. (1pt) Make a scatterplot matrix of the data, with species mapped to colour. Which variables would you expect to be the most important for distinguishing between the species? Is it fair to assume homogeneous variance-covariances? ?????

According to the scatter plot, we should select the plot when 2 area are tend to be more close, also within each group, the data should show a great correlation with each other.

# # A tibble: 6 x 5
#   species     bl    bd     fl     bm
#   <fct>    <dbl> <dbl>  <dbl>  <dbl>
# 1 Adelie  -0.693 0.926 -1.41  -0.680
# 2 Adelie  -0.616 0.280 -1.08  -0.620
# 3 Adelie  -0.462 0.578 -0.477 -1.28 
# 4 Adelie  -1.16  1.22  -0.610 -1.04 
# 5 Adelie  -0.655 1.87  -0.809 -0.799
# 6 Adelie  -0.732 0.479 -1.41  -0.829
# [1] 274   5

# Levene's Test for Homogeneity of Variance (center = median)
#        Df F value Pr(>F)
# group   1  1.6441 0.2009
#       272
# Levene's Test for Homogeneity of Variance (center = median)
#        Df F value  Pr(>F)  
# group   1  3.5213 0.06166 .
#       272                  
# ---
# Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Levene's Test for Homogeneity of Variance (center = median)
#        Df F value Pr(>F)
# group   1   0.157 0.6922
#       272
# Levene's Test for Homogeneity of Variance (center = median)
#        Df F value Pr(>F)
# group   1   1.993 0.1592
#       272

Here, by observing the quantile performance, we can get the bl and fl separate 2 species well.

Then we can get more detailed look of by applying “featurePlot” analysis to have a look at the quantitative variable across the 2 species.

We clearly observe that median and interquartile ranges of the 2 penguin species are quite separated , though by different features in each species. By separation, the means and interquartile range of one class is distinct and don’t overlap with the other classes.

Here the variable bl and fl are estimated to separate the data well.

b. (1pt) Break the data into training and test sets. Fit an LDA model to the data, assuming equal prior probabilities for the two groups. Report confusion table and the misclassification error for the test set.

# Call:
# lda(species ~ ., data = penguin_train, prior = c(0.5, 0.5))
# 
# Prior probabilities of groups:
# Adelie Gentoo 
#    0.5    0.5 
# 
# Group means:
#                bl         bd         fl         bm
# Adelie -0.7748322  0.7029637 -0.8448656 -0.7544736
# Gentoo  0.9019430 -0.9331506  0.9727739  0.8715777
# 
# Coefficients of linear discriminants:
#           LD1
# bl  0.6885892
# bd -1.9509187
# fl  1.4069513
# bm  0.9037988

confusion matrix

# Confusion Matrix and Statistics
# 
#           Reference
# Prediction Adelie Gentoo
#     Adelie     50      0
#     Gentoo      0     41
#                                      
#                Accuracy : 1          
#                  95% CI : (0.9603, 1)
#     No Information Rate : 0.5495     
#     P-Value [Acc > NIR] : < 2.2e-16  
#                                      
#                   Kappa : 1          
#                                      
#  Mcnemar's Test P-Value : NA         
#                                      
#             Sensitivity : 1.0000     
#             Specificity : 1.0000     
#          Pos Pred Value : 1.0000     
#          Neg Pred Value : 1.0000     
#              Prevalence : 0.5495     
#          Detection Rate : 0.5495     
#    Detection Prevalence : 0.5495     
#       Balanced Accuracy : 1.0000     
#                                      
#        'Positive' Class : Adelie     
# 

Here the model appears to be accurate. THe overall accuracy is close to 1 and nearly 1 kappa on the test data.

According to the calculation method of the model error.

We can get that in the current dataset, everything matched well:

# [1] 0

c. (1pt) Write down the LDA rule, for classifying the two species, explicitly being clear about which species is class 1 and which is class 2. ?????

According to the LDA rule we can get that the following equation.New observation of X0 us belongs to class 1 if

In LDA’s definition, Class 1 and 2 need to be mapped in the your data. The class to the right on the reduced dimension will be class 1 in the equqation. So here after comparing the value with Adelie Penguin and Gentoo Penguin

#              LD1
# Adelie -3.775543
# Gentoo  4.597946
# # A tibble: 4 x 2
#   variables `Different mean between group -`
#   <chr>                                <dbl>
# 1 BL                                   -1.68
# 2 BD                                    1.64
# 3 FL                                   -1.82
# 4 BM                                   -1.63
# # A tibble: 4 x 2
#   variables `Different mean between group +`
#   <chr>                                <dbl>
# 1 BL                                   0.127
# 2 BD                                  -0.230
# 3 FL                                   0.128
# 4 BM                                   0.117

d. (1pt) Report the group means and the pooled variance-covariance matrix, and show the computations to obtain the linear discriminant space is computed from these. (Hint: You can use the matlib package to compute matrix inverses, see help here.)

We select the group means:

#                bl         bd         fl         bm
# Adelie -0.7748322  0.7029637 -0.8448656 -0.7544736
# Gentoo  0.9019430 -0.9331506  0.9727739  0.8715777

Here we calculate the variance for each group

The variance of group Adelie

#            bl         bd         fl         bm
# bl 0.22524992 0.12158571 0.06008633 0.12070965
# bd 0.12158571 0.37552815 0.07290338 0.18188810
# fl 0.06008633 0.07290338 0.17334046 0.09076054
# bm 0.12070965 0.18188810 0.09076054 0.25946180

The variance of group Gentoo

#           bl        bd        fl        bm
# bl 0.3423178 0.1845119 0.1516292 0.2441749
# bd 0.1845119 0.2589256 0.1462688 0.2292669
# fl 0.1516292 0.1462688 0.1613951 0.1782303
# bm 0.2441749 0.2292669 0.1782303 0.3879481

Then we got the sum numbers of Adelie and Gentoo which are

According to the formula, we substitute the sum of numbers above into the pooled variance calculation we can get that

Here is the pooled variance-covariance matrix

#          [,1]
# bl  -5.765894
# bd  16.335996
# fl -11.781091
# bm  -7.567949

e. (1pt) Make a plot showing the data in the discriminant space, to examine how well the species are separated.

now we make a plot about the discriminant space of setA.

#    species         bl                 bd                 fl          
#  Adelie:50   Min.   :-2.04076   Min.   :-1.65696   Min.   :-1.73970  
#  Gentoo:41   1st Qu.:-0.78007   1st Qu.:-0.91195   1st Qu.:-0.80934  
#              Median :-0.01981   Median : 0.13106   Median :-0.07834  
#              Mean   : 0.04724   Mean   : 0.06065   Mean   : 0.06114  
#              3rd Qu.: 0.76933   3rd Qu.: 0.87606   3rd Qu.: 0.95170  
#              Max.   : 2.54007   Max.   : 2.16740   Max.   : 1.84884  
#        bm                LD1        
#  Min.   :-1.75620   Min.   :-6.720  
#  1st Qu.:-0.85900   1st Qu.:-4.298  
#  Median : 0.06811   Median :-2.680  
#  Mean   : 0.05201   Mean   :-0.364  
#  3rd Qu.: 0.81578   3rd Qu.: 4.261  
#  Max.   : 1.95223   Max.   : 6.300

Here, by using an density function we can also get that

According to the density plot aren’t intersect with each other, that means there is a good separation of the 2 classes of the Linear Discriminant method.

f. (1pt) Return to the data with the original three species of penguins. Conduct LDA for the three groups, and obtain the 2D discriminant space, using just the same four physical measurements. Using the tour, determine which of the variables are most important for distinguishing Chinstrap penguins from Adelie. Provide a plot or two to support your decision.

# # A tibble: 6 x 5
#   species     bl    bd     fl     bm
#   <fct>    <dbl> <dbl>  <dbl>  <dbl>
# 1 Adelie  -0.883 0.784 -1.42  -0.563
# 2 Adelie  -0.810 0.126 -1.06  -0.501
# 3 Adelie  -0.663 0.430 -0.421 -1.19 
# 4 Adelie  -1.32  1.09  -0.563 -0.937
# 5 Adelie  -0.847 1.75  -0.776 -0.688
# 6 Adelie  -0.920 0.329 -1.42  -0.719
# [1] 342   5

** Box plot and density plot can clearly show the relationship between four variables and easily tell which variable are important for distinguishing**

TOur graph with 4 different dimensions.

# parsnip model object
# 
# Fit time:  20ms 
# Call:
# lda(species ~ ., data = data, prior = ~c(1/3, 1/3, 1/3))
# 
# Prior probabilities of groups:
#    Adelie Chinstrap    Gentoo 
# 0.3333333 0.3333333 0.3333333 
# 
# Group means:
#           bill_length_mm bill_depth_mm flipper_length_mm body_mass_g
# Adelie          38.85545      18.40099          189.6238    3729.455
# Chinstrap       48.46957      18.34783          195.6522    3688.043
# Gentoo          47.67073      14.96707          217.1220    5049.085
# 
# Coefficients of linear discriminants:
#                            LD1          LD2
# bill_length_mm     0.037062318 -0.413284602
# bill_depth_mm     -1.020639811  0.092714992
# flipper_length_mm  0.092626372 -0.002792050
# body_mass_g        0.001515876  0.001782931
# 
# Proportion of trace:
#   LD1   LD2 
# 0.835 0.165

Here we can conclude after taking ‘chinstrap’ into consideration, we can get that bd are easy to tell the difference in order to separate Gentoo from Adelie and Chinstrap. Moreover, the value of bm can also separating Gentoo from Adelie and Chinstrap.

Furthermore, in such situation, we should always foruc on the tour to choose which variable are useful to separate species between Chinstrap and Adelie. According to the tour above, the green point is Adelie , whileas the orange point is chinstrap.According to the tour we can easily tell the bl is the most important criteria for the reason that after the projection, majority of green and orange cannot project on the bl. Based on this witness, we can say bl will retain important data for Chinstrap

a. (0.5pt) Read in the spam data, set the levels for day of the week, and filter to the five most common domains, “com”, “edu”, “net”, “org”, “gov”. Drop variable spampct, because there are many missing values. (This variable was one that some of the mail software provided as a probability that the email was spam. Most mail software didn’t provide this.) Class 1 for this data would be spam=no. Explain why.

According to the rule of thumb, put the most important classes among the 2 classes into class one. For the reason the program asked us about separating spam, so ‘spam=no’ is more important, thus we put it into class 1.

# # A tibble: 1,998 x 20
#    isuid    id `day of week` `time of day` size.kb box   domain local digits
#    <dbl> <dbl> <fct>                 <dbl>   <dbl> <chr> <chr>  <chr>  <dbl>
#  1     1     1 Thu                       0       7 no    com    no         0
#  2     1     2 Thu                       0       2 no    com    no         0
#  3     1     3 Thu                      14       3 no    edu    yes        0
#  4     1     9 Thu                       6       3 yes   edu    yes        0
#  5     1    11 Thu                       7       3 no    com    no         0
#  6     1    13 Thu                       8      12 yes   com    no         0
#  7     1    14 Thu                       8      12 yes   com    no         0
#  8     1    16 Thu                       9       2 yes   edu    yes        0
#  9     1    19 Thu                      10       2 no    edu    yes        0
# 10     1    23 Thu                      12       3 no    edu    yes        0
# # ... with 1,988 more rows, and 11 more variables: name <chr>, cappct <dbl>,
# #   special <dbl>, credit <chr>, sucker <chr>, porn <chr>, chain <chr>,
# #   username <chr>, large text <chr>, category <chr>, spam <chr>

#       [,1] [,2]
#  [1,]    1    2
#  [2,]    1    3
#  [3,]    1    4
#  [4,]    1    5
#  [5,]    1    2
#  [6,]    2    3
#  [7,]    2    4
#  [8,]    2    5
#  [9,]    1    3
# [10,]    2    3
# [11,]    3    4
# [12,]    3    5
# [13,]    1    4
# [14,]    2    4
# [15,]    3    4
# [16,]    4    5
# [17,]    1    5
# [18,]    2    5
# [19,]    3    5
# [20,]    4    5

# [1] 807
# [1] 1998

# [[1]]
# [1] "edu" "net" "org" "gov"
# 
# [[2]]
# [1] "com" "net" "org" "gov"
# 
# [[3]]
# [1] "com" "edu" "org" "gov"
# 
# [[4]]
# [1] "com" "edu" "net" "gov"
# 
# [[5]]
# [1] "com" "edu" "net" "org"
# 
# [[6]]
# [1] "net" "org" "gov"
# 
# [[7]]
# [1] "edu" "org" "gov"
# 
# [[8]]
# [1] "edu" "net" "gov"
# 
# [[9]]
# [1] "edu" "net" "org"
# 
# [[10]]
# [1] "com" "org" "gov"
# 
# [[11]]
# [1] "com" "net" "gov"
# 
# [[12]]
# [1] "com" "net" "org"
# 
# [[13]]
# [1] "com" "edu" "gov"
# 
# [[14]]
# [1] "com" "edu" "org"
# 
# [[15]]
# [1] "com" "edu" "net"

# [[1]]
# [1] "com"
# 
# [[2]]
# [1] "edu"
# 
# [[3]]
# [1] "net"
# 
# [[4]]
# [1] "org"
# 
# [[5]]
# [1] "gov"
# 
# [[6]]
# [1] "com" "edu"
# 
# [[7]]
# [1] "com" "net"
# 
# [[8]]
# [1] "com" "org"
# 
# [[9]]
# [1] "com" "gov"
# 
# [[10]]
# [1] "edu" "net"
# 
# [[11]]
# [1] "edu" "org"
# 
# [[12]]
# [1] "edu" "gov"
# 
# [[13]]
# [1] "net" "org"
# 
# [[14]]
# [1] "net" "gov"
# 
# [[15]]
# [1] "org" "gov"

# # A tibble: 15 x 2
#    index     ose
#    <dbl>   <dbl>
#  1     1 0.694  
#  2     2 0.00868
#  3     3 0.729  
#  4     4 0.154  
#  5     5 0.1    
#  6     6 0.309  
#  7     7 0.698  
#  8     8 0.677  
#  9     9 0.687  
# 10    10 0.0823 
# 11    11 0.0122 
# 12    12 0.00955
# 13    13 0.625  
# 14    14 0.680  
# 15    15 0.139

#      optimal_ose optimal_ose_v
# [1,]           0           113

#  [1] "isuid"       "id"          "day of week" "time of day" "size.kb"    
#  [6] "box"         "domain"      "local"       "digits"      "name"       
# [11] "cappct"      "special"     "credit"      "sucker"      "porn"       
# [16] "chain"       "username"    "large text"  "category"    "spam"

#      optimal_ose optimal_ose_v
# [1,]   0.2776457             7

#      optimal_ose optimal_ose_v
# [1,]   0.2638581             0

#      optimal_ose optimal_ose_v
# [1,]   0.2657224             0

#      optimal_ose optimal_ose_v
# [1,]    0.273913             0

# spec_tbl_df[,10] [167 x 10] (S3: spec_tbl_df/tbl_df/tbl/data.frame)
#  $ country   : chr [1:167] "Afghanistan" "Albania" "Algeria" "Angola" ...
#  $ child_mort: num [1:167] 90.2 16.6 27.3 119 10.3 14.5 18.1 4.8 4.3 39.2 ...
#  $ exports   : num [1:167] 10 28 38.4 62.3 45.5 18.9 20.8 19.8 51.3 54.3 ...
#  $ health    : num [1:167] 7.58 6.55 4.17 2.85 6.03 8.1 4.4 8.73 11 5.88 ...
#  $ imports   : num [1:167] 44.9 48.6 31.4 42.9 58.9 16 45.3 20.9 47.8 20.7 ...
#  $ income    : num [1:167] 1610 9930 12900 5900 19100 18700 6700 41400 43200 16000 ...
#  $ inflation : num [1:167] 9.44 4.49 16.1 22.4 1.44 20.9 7.77 1.16 0.873 13.8 ...
#  $ life_expec: num [1:167] 56.2 76.3 76.5 60.1 76.8 75.8 73.3 82 80.5 69.1 ...
#  $ total_fer : num [1:167] 5.82 1.65 2.89 6.16 2.13 2.37 1.69 1.93 1.44 1.92 ...
#  $ gdpp      : num [1:167] 553 4090 4460 3530 12200 10300 3220 51900 46900 5840 ...
#  - attr(*, "spec")=
#   .. cols(
#   ..   country = col_character(),
#   ..   child_mort = col_double(),
#   ..   exports = col_double(),
#   ..   health = col_double(),
#   ..   imports = col_double(),
#   ..   income = col_double(),
#   ..   inflation = col_double(),
#   ..   life_expec = col_double(),
#   ..   total_fer = col_double(),
#   ..   gdpp = col_double()
#   .. )

# Standard deviations (1, .., p=9):
# [1] 2.0336314 1.2435217 1.0818425 0.9973889 0.8127847 0.4728437 0.3368067
# [8] 0.2971790 0.2586020
# 
# Rotation (n x k) = (9 x 9):
#                   PC1          PC2         PC3          PC4         PC5
# child_mort -0.4195194 -0.192883937  0.02954353 -0.370653262  0.16896968
# exports     0.2838970 -0.613163494 -0.14476069 -0.003091019 -0.05761584
# health      0.1508378  0.243086779  0.59663237 -0.461897497 -0.51800037
# imports     0.1614824 -0.671820644  0.29992674  0.071907461 -0.25537642
# income      0.3984411 -0.022535530 -0.30154750 -0.392159039  0.24714960
# inflation  -0.1931729  0.008404473 -0.64251951 -0.150441762 -0.71486910
# life_expec  0.4258394  0.222706743 -0.11391854  0.203797235 -0.10821980
# total_fer  -0.4037290 -0.155233106 -0.01954925 -0.378303645  0.13526221
# gdpp        0.3926448  0.046022396 -0.12297749 -0.531994575  0.18016662
#                     PC6         PC7         PC8         PC9
# child_mort -0.200628153  0.07948854  0.68274306  0.32754180
# exports     0.059332832  0.70730269  0.01419742 -0.12308207
# health     -0.007276456  0.24983051 -0.07249683  0.11308797
# imports     0.030031537 -0.59218953  0.02894642  0.09903717
# income     -0.160346990 -0.09556237 -0.35262369  0.61298247
# inflation  -0.066285372 -0.10463252  0.01153775 -0.02523614
# life_expec  0.601126516 -0.01848639  0.50466425  0.29403981
# total_fer   0.750688748 -0.02882643 -0.29335267 -0.02633585
# gdpp       -0.016778761 -0.24299776  0.24969636 -0.62564572

# Importance of components:
#                           PC1    PC2    PC3    PC4    PC5     PC6    PC7
# Standard deviation     2.0336 1.2435 1.0818 0.9974 0.8128 0.47284 0.3368
# Proportion of Variance 0.4595 0.1718 0.1300 0.1105 0.0734 0.02484 0.0126
# Cumulative Proportion  0.4595 0.6313 0.7614 0.8719 0.9453 0.97015 0.9828
#                            PC8     PC9
# Standard deviation     0.29718 0.25860
# Proportion of Variance 0.00981 0.00743
# Cumulative Proportion  0.99257 1.00000

# # A tibble: 9 x 9
#      PC1      PC2     PC3      PC4     PC5      PC6     PC7     PC8     PC9
#    <dbl>    <dbl>   <dbl>    <dbl>   <dbl>    <dbl>   <dbl>   <dbl>   <dbl>
# 1  0.426  0.223   -0.114   0.204   -0.108   0.601   -0.0185  0.505   0.294 
# 2  0.398 -0.0225  -0.302  -0.392    0.247  -0.160   -0.0956 -0.353   0.613 
# 3  0.393  0.0460  -0.123  -0.532    0.180  -0.0168  -0.243   0.250  -0.626 
# 4  0.284 -0.613   -0.145  -0.00309 -0.0576  0.0593   0.707   0.0142 -0.123 
# 5  0.161 -0.672    0.300   0.0719  -0.255   0.0300  -0.592   0.0289  0.0990
# 6  0.151  0.243    0.597  -0.462   -0.518  -0.00728  0.250  -0.0725  0.113 
# 7 -0.193  0.00840 -0.643  -0.150   -0.715  -0.0663  -0.105   0.0115 -0.0252
# 8 -0.404 -0.155   -0.0195 -0.378    0.135   0.751   -0.0288 -0.293  -0.0263
# 9 -0.420 -0.193    0.0295 -0.371    0.169  -0.201    0.0795  0.683   0.328

# Standard deviations (1, .., p=5):
# [1] 1.9065314 0.9721904 0.4821562 0.3239670 0.2873232
# 
# Rotation (n x k) = (5 x 5):
#                   PC1        PC2           PC3        PC4        PC5
# child_mort -0.4654859 -0.4031338  1.944752e-01 -0.2900511  0.7062972
# income      0.4297577 -0.5358646  1.499573e-01  0.6779796  0.2145085
# life_expec  0.4790410  0.2223537 -6.274361e-01 -0.1626967  0.5485730
# total_fer  -0.4420473 -0.4036622 -7.389291e-01  0.1993868 -0.2363890
# gdpp        0.4168274 -0.5813329  1.490296e-05 -0.6244907 -0.3135575

# Importance of components:
#                          PC1    PC2     PC3     PC4     PC5
# Standard deviation     1.907 0.9722 0.48216 0.32397 0.28732
# Proportion of Variance 0.727 0.1890 0.04649 0.02099 0.01651
# Cumulative Proportion  0.727 0.9160 0.96250 0.98349 1.00000

Citation

e1071 David Meyer, Evgenia Dimitriadou, Kurt Hornik, Andreas Weingessel and Friedrich Leisch (2021). e1071: Misc Functions of the Department of Statistics, Probability Theory Group (Formerly: E1071), TU Wien. R package version 1.7-6. https://CRAN.R-project.org/package=e1071

tidyverse Wickham et al., (2019). Welcome to the tidyverse. Journal of Open Source Software, 4(43), 1686, https://doi.org/10.21105/joss.01686

GGally Barret Schloerke, Di Cook, Joseph Larmarange, Francois Briatte, Moritz Marbach, Edwin Thoen, Amos Elberg and Jason Crowley (2021). GGally: Extension to ‘ggplot2’. R package version 2.1.1. https://CRAN.R-project.org/package=GGally

MASS Venables, W. N. & Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth Edition. Springer, New York. ISBN 0-387-95457-0

Discrim

Max Kuhn (2020). discrim: Model Wrappers for Discriminant Analysis. R package version 0.1.1. https://CRAN.R-project.org/package=discrim

tidymodels

Kuhn et al., (2020). Tidymodels: a collection of packages for modeling and machine learning using tidyverse principles. https://www.tidymodels.org

KableExtra Hao Zhu (2021). kableExtra: Construct Complex Table with ‘kable’ and Pipe Syntax. R package version 1.3.4. https://CRAN.R-project.org/package=kableExtra

rpart

Terry Therneau and Beth Atkinson (2019). rpart: Recursive Partitioning and Regression Trees. R package version 4.1-15. https://CRAN.R-project.org/package=rpart

palmerpenguins

Horst AM, Hill AP, Gorman KB (2020). palmerpenguins: Palmer Archipelago (Antarctica) penguin data. R package version 0.1.0. https://allisonhorst.github.io/palmerpenguins/

plotly

C. Sievert. Interactive Web-Based Data Visualization with R, plotly, and shiny. Chapman and Hall/CRC Florida, 2020.

tidyr

Hadley Wickham (2021). tidyr: Tidy Messy Data. R package version 1.1.3. https://CRAN.R-project.org/package=tidyr

tourr

Hadley Wickham, Dianne Cook, Heike Hofmann, Andreas Buja (2011). tourr: An R Package for Exploring Multivariate Data with Projections. Journal of Statistical Software, 40(2), 1-18. URL http://www.jstatsoft.org/v40/i02/.

rsample Julia Silge, Fanny Chow, Max Kuhn and Hadley Wickham (2021). rsample: General Resampling Infrastructure. R package version 0.0.9. https://CRAN.R-project.org/package=rsample

persnip

Max Kuhn and Davis Vaughan (2021). parsnip: A Common API to Modeling and Analysis Functions. R package version 0.1.5. https://CRAN.R-project.org/package=parsnip

yardstick

Max Kuhn and Davis Vaughan (2021). yardstick: Tidy Characterizations of Model Performance. R package version 0.0.8. https://CRAN.R-project.org/package=yardstick

spinifex

Nicholas Spyrison and Dianne Cook (2021). spinifex: Manual Tours, Manual Control of Dynamic Projections of Numeric Multivariate Data. R package version 0.2.8. https://CRAN.R-project.org/package=spinifex

dplyr

Hadley Wickham, Romain François, Lionel Henry and Kirill Müller (2021). dplyr: A Grammar of Data Manipulation. R package version 1.0.5. https://CRAN.R-project.org/package=dplyr

skimr Elin Waring, Michael Quinn, Amelia McNamara, Eduardo Arino de la Rubia, Hao Zhu and Shannon Ellis (2021). skimr: Compact and Flexible Summaries of Data. R package version 2.1.3. https://CRAN.R-project.org/package=skimr

caret Max Kuhn (2020). caret: Classification and Regression Training. R package version 6.0-86. https://CRAN.R-project.org/package=caret

ggplot2 H. Wickham. ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag New York, 2016.

klaR Weihs, C., Ligges, U., Luebke, K. and Raabe, N. (2005). klaR Analyzing German Business Cycles. In Baier, D., Decker, R. and Schmidt-Thieme, L. (eds.). Data Analysis and Decision Support, 335-343, Springer-Verlag, Berlin.

knitr

Yihui Xie (2021). knitr: A General-Purpose Package for Dynamic Report Generation in R. R package version 1.33.

Yihui Xie (2015) Dynamic Documents with R and knitr. 2nd edition. Chapman and Hall/CRC. ISBN 978-1498716963

Yihui Xie (2014) knitr: A Comprehensive Tool for Reproducible Research in R. In Victoria Stodden, Friedrich Leisch and Roger D. Peng, editors, Implementing Reproducible Computational Research. Chapman and Hall/CRC. ISBN 978-1466561595

boot Angelo Canty and Brian Ripley (2021). boot: Bootstrap R (S-Plus) Functions. R package version 1.3-27.

Davison, A. C. & Hinkley, D. V. (1997) Bootstrap Methods and Their Applications. Cambridge University Press, Cambridge. ISBN 0-521-57391-2

ggrepel Kamil Slowikowski (2021). ggrepel: Automatically Position Non-Overlapping Text Labels with ‘ggplot2’. R package version 0.9.1. https://CRAN.R-project.org/package=ggrepel

# 
# To cite package 'ggrepel' in publications use:
# 
#   Kamil Slowikowski (2021). ggrepel: Automatically Position
#   Non-Overlapping Text Labels with 'ggplot2'. R package version 0.9.1.
#   https://CRAN.R-project.org/package=ggrepel
# 
# A BibTeX entry for LaTeX users is
# 
#   @Manual{,
#     title = {ggrepel: Automatically Position Non-Overlapping Text Labels with
# 'ggplot2'},
#     author = {Kamil Slowikowski},
#     year = {2021},
#     note = {R package version 0.9.1},
#     url = {https://CRAN.R-project.org/package=ggrepel},
#   }
# 
#   Wickham et al., (2019). Welcome to the tidyverse. Journal of Open
#   Source Software, 4(43), 1686, https://doi.org/10.21105/joss.01686
# 
# A BibTeX entry for LaTeX users is
# 
#   @Article{,
#     title = {Welcome to the {tidyverse}},
#     author = {Hadley Wickham and Mara Averick and Jennifer Bryan and Winston Chang and Lucy D'Agostino McGowan and Romain François and Garrett Grolemund and Alex Hayes and Lionel Henry and Jim Hester and Max Kuhn and Thomas Lin Pedersen and Evan Miller and Stephan Milton Bache and Kirill Müller and Jeroen Ooms and David Robinson and Dana Paige Seidel and Vitalie Spinu and Kohske Takahashi and Davis Vaughan and Claus Wilke and Kara Woo and Hiroaki Yutani},
#     year = {2019},
#     journal = {Journal of Open Source Software},
#     volume = {4},
#     number = {43},
#     pages = {1686},
#     doi = {10.21105/joss.01686},
#   }
# 
#   Kuhn et al., (2020). Tidymodels: a collection of packages for
#   modeling and machine learning using tidyverse principles.
#   https://www.tidymodels.org
# 
# A BibTeX entry for LaTeX users is
# 
#   @Manual{,
#     title = {Tidymodels: a collection of packages for modeling and machine learning using tidyverse principles.},
#     author = {Max Kuhn and Hadley Wickham},
#     url = {https://www.tidymodels.org},
#     year = {2020},
#   }
# 
# To cite package 'discrim' in publications use:
# 
#   Max Kuhn (2020). discrim: Model Wrappers for Discriminant Analysis. R
#   package version 0.1.1. https://CRAN.R-project.org/package=discrim
# 
# A BibTeX entry for LaTeX users is
# 
#   @Manual{,
#     title = {discrim: Model Wrappers for Discriminant Analysis},
#     author = {Max Kuhn},
#     year = {2020},
#     note = {R package version 0.1.1},
#     url = {https://CRAN.R-project.org/package=discrim},
#   }
# 
# To cite the MASS package in publications use:
# 
#   Venables, W. N. & Ripley, B. D. (2002) Modern Applied Statistics with
#   S. Fourth Edition. Springer, New York. ISBN 0-387-95457-0
# 
# A BibTeX entry for LaTeX users is
# 
#   @Book{,
#     title = {Modern Applied Statistics with S},
#     author = {W. N. Venables and B. D. Ripley},
#     publisher = {Springer},
#     edition = {Fourth},
#     address = {New York},
#     year = {2002},
#     note = {ISBN 0-387-95457-0},
#     url = {https://www.stats.ox.ac.uk/pub/MASS4/},
#   }
# 
# To cite the 'boot' package in publications use:
# 
#   Angelo Canty and Brian Ripley (2021). boot: Bootstrap R (S-Plus)
#   Functions. R package version 1.3-27.
# 
#   Davison, A. C. & Hinkley, D. V. (1997) Bootstrap Methods and Their
#   Applications. Cambridge University Press, Cambridge. ISBN
#   0-521-57391-2
# 
# To see these entries in BibTeX format, use 'print(<citation>,
# bibtex=TRUE)', 'toBibtex(.)', or set
# 'options(citation.bibtex.max=999)'.
# 
# To cite package 'ggrepel' in publications use:
# 
#   Kamil Slowikowski (2021). ggrepel: Automatically Position
#   Non-Overlapping Text Labels with 'ggplot2'. R package version 0.9.1.
#   https://CRAN.R-project.org/package=ggrepel
# 
# A BibTeX entry for LaTeX users is
# 
#   @Manual{,
#     title = {ggrepel: Automatically Position Non-Overlapping Text Labels with
# 'ggplot2'},
#     author = {Kamil Slowikowski},
#     year = {2021},
#     note = {R package version 0.9.1},
#     url = {https://CRAN.R-project.org/package=ggrepel},
#   }
# 
# To cite package 'kableExtra' in publications use:
# 
#   Hao Zhu (2021). kableExtra: Construct Complex Table with 'kable' and
#   Pipe Syntax. R package version 1.3.4.
#   https://CRAN.R-project.org/package=kableExtra
# 
# A BibTeX entry for LaTeX users is
# 
#   @Manual{,
#     title = {kableExtra: Construct Complex Table with 'kable' and Pipe Syntax},
#     author = {Hao Zhu},
#     year = {2021},
#     note = {R package version 1.3.4},
#     url = {https://CRAN.R-project.org/package=kableExtra},
#   }
# 
# To cite package 'caret' in publications use:
# 
#   Max Kuhn (2020). caret: Classification and Regression Training. R
#   package version 6.0-86. https://CRAN.R-project.org/package=caret
# 
# A BibTeX entry for LaTeX users is
# 
#   @Manual{,
#     title = {caret: Classification and Regression Training},
#     author = {Max Kuhn},
#     year = {2020},
#     note = {R package version 6.0-86},
#     url = {https://CRAN.R-project.org/package=caret},
#   }
# 
# To cite package 'dplyr' in publications use:
# 
#   Hadley Wickham, Romain François, Lionel Henry and Kirill Müller
#   (2021). dplyr: A Grammar of Data Manipulation. R package version
#   1.0.5. https://CRAN.R-project.org/package=dplyr
# 
# A BibTeX entry for LaTeX users is
# 
#   @Manual{,
#     title = {dplyr: A Grammar of Data Manipulation},
#     author = {Hadley Wickham and Romain François and Lionel Henry and Kirill Müller},
#     year = {2021},
#     note = {R package version 1.0.5},
#     url = {https://CRAN.R-project.org/package=dplyr},
#   }
# 
# To cite tourr in publications use:
# 
#   Hadley Wickham, Dianne Cook, Heike Hofmann, Andreas Buja (2011).
#   tourr: An R Package for Exploring Multivariate Data with Projections.
#   Journal of Statistical Software, 40(2), 1-18. URL
#   http://www.jstatsoft.org/v40/i02/.
# 
# When using the slice tour, please cite:
# 
#   Ursula Laa, Dianne Cook, German Valencia (2020). A slice tour for
#   finding hollowness in high-dimensional data. Journal of Computational
#   and Graphical Statistics, 29:3, 681-687. DOI:
#   10.1080/10618600.2020.1777140
# 
# To see these entries in BibTeX format, use 'print(<citation>,
# bibtex=TRUE)', 'toBibtex(.)', or set
# 'options(citation.bibtex.max=999)'.
# 
# To cite package 'magrittr' in publications use:
# 
#   Stefan Milton Bache and Hadley Wickham (2020). magrittr: A
#   Forward-Pipe Operator for R. R package version 2.0.1.
#   https://CRAN.R-project.org/package=magrittr
# 
# A BibTeX entry for LaTeX users is
# 
#   @Manual{,
#     title = {magrittr: A Forward-Pipe Operator for R},
#     author = {Stefan Milton Bache and Hadley Wickham},
#     year = {2020},
#     note = {R package version 2.0.1},
#     url = {https://CRAN.R-project.org/package=magrittr},
#   }
# 
# To cite ggplot2 in publications, please use:
# 
#   H. Wickham. ggplot2: Elegant Graphics for Data Analysis.
#   Springer-Verlag New York, 2016.
# 
# A BibTeX entry for LaTeX users is
# 
#   @Book{,
#     author = {Hadley Wickham},
#     title = {ggplot2: Elegant Graphics for Data Analysis},
#     publisher = {Springer-Verlag New York},
#     year = {2016},
#     isbn = {978-3-319-24277-4},
#     url = {https://ggplot2.tidyverse.org},
#   }
# 
# To cite package 'rpart' in publications use:
# 
#   Terry Therneau and Beth Atkinson (2019). rpart: Recursive
#   Partitioning and Regression Trees. R package version 4.1-15.
#   https://CRAN.R-project.org/package=rpart
# 
# A BibTeX entry for LaTeX users is
# 
#   @Manual{,
#     title = {rpart: Recursive Partitioning and Regression Trees},
#     author = {Terry Therneau and Beth Atkinson},
#     year = {2019},
#     note = {R package version 4.1-15},
#     url = {https://CRAN.R-project.org/package=rpart},
#   }
# 
# To cite package 'rpart.plot' in publications use:
# 
#   Stephen Milborrow (2020). rpart.plot: Plot 'rpart' Models: An
#   Enhanced Version of 'plot.rpart'. R package version 3.0.9.
#   https://CRAN.R-project.org/package=rpart.plot
# 
# A BibTeX entry for LaTeX users is
# 
#   @Manual{,
#     title = {rpart.plot: Plot 'rpart' Models: An Enhanced Version of 'plot.rpart'},
#     author = {Stephen Milborrow},
#     year = {2020},
#     note = {R package version 3.0.9},
#     url = {https://CRAN.R-project.org/package=rpart.plot},
#   }
# 
# ATTENTION: This citation information has been auto-generated from the
# package DESCRIPTION file and may need manual editing, see
# 'help("citation")'.
# 
# To cite the 'knitr' package in publications use:
# 
#   Yihui Xie (2021). knitr: A General-Purpose Package for Dynamic Report
#   Generation in R. R package version 1.33.
# 
#   Yihui Xie (2015) Dynamic Documents with R and knitr. 2nd edition.
#   Chapman and Hall/CRC. ISBN 978-1498716963
# 
#   Yihui Xie (2014) knitr: A Comprehensive Tool for Reproducible
#   Research in R. In Victoria Stodden, Friedrich Leisch and Roger D.
#   Peng, editors, Implementing Reproducible Computational Research.
#   Chapman and Hall/CRC. ISBN 978-1466561595
# 
# To see these entries in BibTeX format, use 'print(<citation>,
# bibtex=TRUE)', 'toBibtex(.)', or set
# 'options(citation.bibtex.max=999)'.
# 
# To cite klaR in publications use:
# 
#   Weihs, C., Ligges, U., Luebke, K. and Raabe, N. (2005). klaR
#   Analyzing German Business Cycles. In Baier, D., Decker, R. and
#   Schmidt-Thieme, L. (eds.). Data Analysis and Decision Support,
#   335-343, Springer-Verlag, Berlin.
# 
# A BibTeX entry for LaTeX users is
# 
#   @InProceedings{,
#     title = {klaR Analyzing German Business Cycles},
#     author = {Claus Weihs and Uwe Ligges and Karsten Luebke and Nils Raabe},
#     booktitle = {Data Analysis and Decision Support},
#     editor = {D. Baier and R. Decker and L. Schmidt-Thieme},
#     publisher = {Springer-Verlag},
#     address = {Berlin},
#     year = {2005},
#     pages = {335-343},
#   }
# 
# To cite package 'skimr' in publications use:
# 
#   Elin Waring, Michael Quinn, Amelia McNamara, Eduardo Arino de la
#   Rubia, Hao Zhu and Shannon Ellis (2021). skimr: Compact and Flexible
#   Summaries of Data. R package version 2.1.3.
#   https://CRAN.R-project.org/package=skimr
# 
# A BibTeX entry for LaTeX users is
# 
#   @Manual{,
#     title = {skimr: Compact and Flexible Summaries of Data},
#     author = {Elin Waring and Michael Quinn and Amelia McNamara and Eduardo {Arino de la Rubia} and Hao Zhu and Shannon Ellis},
#     year = {2021},
#     note = {R package version 2.1.3},
#     url = {https://CRAN.R-project.org/package=skimr},
#   }
# 
# To cite package 'spinifex' in publications use:
# 
#   Nicholas Spyrison and Dianne Cook (2021). spinifex: Manual Tours,
#   Manual Control of Dynamic Projections of Numeric Multivariate Data. R
#   package version 0.2.8. https://CRAN.R-project.org/package=spinifex
# 
# A BibTeX entry for LaTeX users is
# 
#   @Manual{,
#     title = {spinifex: Manual Tours, Manual Control of Dynamic Projections of Numeric
# Multivariate Data},
#     author = {Nicholas Spyrison and Dianne Cook},
#     year = {2021},
#     note = {R package version 0.2.8},
#     url = {https://CRAN.R-project.org/package=spinifex},
#   }
# 
# To cite package 'yardstick' in publications use:
# 
#   Max Kuhn and Davis Vaughan (2021). yardstick: Tidy Characterizations
#   of Model Performance. R package version 0.0.8.
#   https://CRAN.R-project.org/package=yardstick
# 
# A BibTeX entry for LaTeX users is
# 
#   @Manual{,
#     title = {yardstick: Tidy Characterizations of Model Performance},
#     author = {Max Kuhn and Davis Vaughan},
#     year = {2021},
#     note = {R package version 0.0.8},
#     url = {https://CRAN.R-project.org/package=yardstick},
#   }
# 
# To cite package 'parsnip' in publications use:
# 
#   Max Kuhn and Davis Vaughan (2021). parsnip: A Common API to Modeling
#   and Analysis Functions. R package version 0.1.5.
#   https://CRAN.R-project.org/package=parsnip
# 
# A BibTeX entry for LaTeX users is
# 
#   @Manual{,
#     title = {parsnip: A Common API to Modeling and Analysis Functions},
#     author = {Max Kuhn and Davis Vaughan},
#     year = {2021},
#     note = {R package version 0.1.5},
#     url = {https://CRAN.R-project.org/package=parsnip},
#   }
# 
# To cite package 'rsample' in publications use:
# 
#   Julia Silge, Fanny Chow, Max Kuhn and Hadley Wickham (2021). rsample:
#   General Resampling Infrastructure. R package version 0.0.9.
#   https://CRAN.R-project.org/package=rsample
# 
# A BibTeX entry for LaTeX users is
# 
#   @Manual{,
#     title = {rsample: General Resampling Infrastructure},
#     author = {Julia Silge and Fanny Chow and Max Kuhn and Hadley Wickham},
#     year = {2021},
#     note = {R package version 0.0.9},
#     url = {https://CRAN.R-project.org/package=rsample},
#   }
# 
# To cite plotly in publications use:
# 
#   C. Sievert. Interactive Web-Based Data Visualization with R, plotly,
#   and shiny. Chapman and Hall/CRC Florida, 2020.
# 
# A BibTeX entry for LaTeX users is
# 
#   @Book{,
#     author = {Carson Sievert},
#     title = {Interactive Web-Based Data Visualization with R, plotly, and shiny},
#     publisher = {Chapman and Hall/CRC},
#     year = {2020},
#     isbn = {9781138331457},
#     url = {https://plotly-r.com},
#   }
# 
# To cite package 'GGally' in publications use:
# 
#   Barret Schloerke, Di Cook, Joseph Larmarange, Francois Briatte,
#   Moritz Marbach, Edwin Thoen, Amos Elberg and Jason Crowley (2021).
#   GGally: Extension to 'ggplot2'. R package version 2.1.1.
#   https://CRAN.R-project.org/package=GGally
# 
# A BibTeX entry for LaTeX users is
# 
#   @Manual{,
#     title = {GGally: Extension to 'ggplot2'},
#     author = {Barret Schloerke and Di Cook and Joseph Larmarange and Francois Briatte and Moritz Marbach and Edwin Thoen and Amos Elberg and Jason Crowley},
#     year = {2021},
#     note = {R package version 2.1.1},
#     url = {https://CRAN.R-project.org/package=GGally},
#   }
# 
# To cite package 'pracma' in publications use:
# 
#   Hans W. Borchers (2021). pracma: Practical Numerical Math Functions.
#   R package version 2.3.3. https://CRAN.R-project.org/package=pracma
# 
# A BibTeX entry for LaTeX users is
# 
#   @Manual{,
#     title = {pracma: Practical Numerical Math Functions},
#     author = {Hans W. Borchers},
#     year = {2021},
#     note = {R package version 2.3.3},
#     url = {https://CRAN.R-project.org/package=pracma},
#   }
# 
# To cite the car package in publications use:
# 
#   John Fox and Sanford Weisberg (2019). An {R} Companion to Applied
#   Regression, Third Edition. Thousand Oaks CA: Sage. URL:
#   https://socialsciences.mcmaster.ca/jfox/Books/Companion/
# 
# A BibTeX entry for LaTeX users is
# 
#   @Book{,
#     title = {An {R} Companion to Applied Regression},
#     edition = {Third},
#     author = {John Fox and Sanford Weisberg},
#     year = {2019},
#     publisher = {Sage},
#     address = {Thousand Oaks {CA}},
#     url = {https://socialsciences.mcmaster.ca/jfox/Books/Companion/},
#   }
# 
# Please cite the 'rattle' package in publications using:
# 
#   Williams, G. J. (2011), Data Mining with Rattle and R: The Art of
#   Excavating Data for Knowledge Discovery, Use R!, Springer.
# 
# A BibTeX entry for LaTeX users is
# 
#   @Book{,
#     title = {Data Mining with {Rattle} and {R}: The art of excavating data for knowledge discovery},
#     author = {Graham J. Williams},
#     publisher = {Springer},
#     series = {Use R!},
#     year = {2011},
#     url = {http://www.amazon.com/gp/product/1441998896/ref=as_li_qf_sp_asin_tl?ie=UTF8&tag=togaware-20&linkCode=as2&camp=217145&creative=399373&creativeASIN=1441998896},
#   }